Move blocks between sheets, maintaining references


I have my schematic broken up into sheets, but have changed my mind about how things are broken up. I would like to move a block from one sheet to another. If I save / paste, all the references change back to R?, U?, multi-unit ICs default to unit A, etc.

I need to move while keeping the references.

I read somewhere that File -> Append Schematic Sheet would accomplish this. I tried it, but they were reset to defaults, same as if I had saved and pasted.

Is there a way do to this?